Math, Statistics & Experimentation

Cosine similarity measures only the angle between two vectors; Euclidean distance also reacts to their magnitudes. Two term-count vectors with the same word mix but different document lengths score cosine 1.0 while sitting far apart in Euclidean distance.

A nonzero vector v is an eigenvector of A when Av = lambda v. Multiplying by A leaves v on its own line through the origin, only stretching, shrinking or flipping it; the scalar lambda is that scale factor.

A zero determinant means the matrix is singular: no inverse exists. For a 2x2 matrix [[a, b], [c, d]] the determinant is ad - bc, so ad - bc = 0 is the exact test for non-invertibility.

Matrix multiplication needs matching inner dimensions: if A is m x n and B is n x p, then AB is m x p. Entry (i,j) is the dot product of row i of A with column j of B.

Run Gaussian elimination on the augmented matrix. A row that is all zeros on the left but nonzero on the right means no solution. Otherwise, one pivot per unknown means exactly one solution; any free unknown means infinitely many.

A set is convex if the straight segment joining any two of its points lies entirely inside it: for every x and y in the set C and every t in [0, 1], the point t*x + (1-t)*y is also in C.

At a smooth interior local minimum the gradient must vanish, so a zero gradient is necessary. But maxima, saddle points and flat inflection points also have a zero gradient, so vanishing slope on its own certifies nothing.

The derivative of x^2 is 2x. The difference quotient ((x+h)^2 - x^2)/h expands to (2xh + h^2)/h, which simplifies to 2x + h, and letting h shrink to 0 leaves 2x.

Each iteration moves every parameter a short distance opposite its own partial derivative: new value = old value minus eta times the gradient. The learning rate eta scales how far you move; the loop repeats until the gradient is near zero.

Differentiate one variable at a time, holding the other fixed: df/dx = 2xy and df/dy = x^2. At (2, 3) those evaluate to 12 and 4, so the gradient there is the vector (12, 4).

Break the target number into a chain of estimable factors: city population, people per household, share of households owning a piano, tunings per piano per year, and tunings one tuner performs per year. Multiply through, divide, and land near 50.

A joint proportion divides a cell count by the grand total. A marginal proportion divides a row or column total by the grand total. A conditional proportion divides a cell by its own row or column total.

A correlation only says the two series move up and down together in the observed data. It cannot say ice cream causes drownings: something else, such as hot weather, can drive both, and correlation carries no direction while causation does.

Sample covariance measures whether two variables move together, but it carries the product of their units, so its size means little alone. Pearson's r divides covariance by both standard deviations, giving a unitless number between -1 and 1.

The mean sums every value, so a long right tail of very high incomes pulls it upward. The median depends only on rank, so extreme values barely move it. Mean above median is the usual signature of right skew.

Your first pick wins only one time in three, so two times in three the car is behind another door. The host, who knows where it is, opens a losing door and concentrates that 2/3 onto the single door left.

The Central Limit Theorem says that averaging many independent draws from almost any distribution with finite variance produces an average whose distribution is approximately normal, even when the individual observations are not remotely normal.

About 2.5%. A score of 130 is two standard deviations above the mean, so its z-score is 2; the rule puts roughly 95% of values within two SDs, leaving about 5% split evenly between the two tails.

A binomial count over n independent trials with success probability p has mean np and variance np(1-p). It is the sum of n Bernoulli(p) indicators, each contributing mean p and variance p(1-p).

The Poisson mass function is P(X = k) = e^(-λ) λ^k / k!. With λ = 3 tickets per hour, P(X = 0) = e^(-3), about 0.0498, so roughly a 5 percent chance of a completely quiet hour.

One-way ANOVA tests whether the population means of three or more groups defined by a single factor are all equal. The null says every group mean is the same; the alternative says at least one differs.

A two-sample t-test assumes observations are independent within and between groups, that the sampling distribution of each group mean is approximately normal, and, for the pooled version, that the two populations share a common variance.

The bootstrap treats your sample as a stand-in for the population: draw many new samples of the same size with replacement, recompute the statistic on each, and read the interval off the middle 95% of those values.

A chi-square goodness-of-fit test compares observed category counts with the counts a hypothesised distribution predicts. A fair die over 600 rolls predicts 100 per face; the statistic sums (observed minus expected) squared, divided by expected, across the six faces.

The 95% describes the procedure, not one interval. If you repeated the sampling and rebuilt the interval many times, about 95% of those intervals would contain the fixed true parameter. Any single interval either covers it or does not.

Readings from one patient are correlated, so three readings carry far less information than three different patients. Counting them as independent inflates the sample size, shrinks the standard errors and p-values, and manufactures significance that is not there.

The standard error of a regression coefficient measures how much that estimate would move around if you refit the model on other samples from the same process. A smaller standard error means a more precisely estimated effect.

The Saturday coefficient is the estimated difference in the outcome between Saturday and Monday, with the model's other predictors held fixed. It is a contrast against the omitted baseline day, not Saturday's own average level.

Record the user as right-censored at 21 days: the subscription lasted at least 21 days, and the eventual cancellation time is unknown. Deleting the row biases survival downward; coding it as a cancellation invents an event that never happened.

R-squared is the share of the outcome's total variation that the fitted model accounts for, computed as 1 minus the residual sum of squares over the total sum of squares. It measures fit on the data used, not correctness.

It is the posterior probability that variant B's true rate is higher than variant A's, given the data and the prior. It is a claim about the unknown rates, and says nothing about how large the difference is.

The frequency reading defines probability as the proportion of times an outcome occurs in repeatable trials. The belief reading defines it as a numeric degree of confidence, so it can also score one-off events such as a single rocket launch.

Beta(1 + 8, 1 + 92), that is Beta(9, 93). The prior contributes one pseudo-success and one pseudo-failure, so the posterior mean is 9/102, about 0.088, slightly above the raw rate of 0.08.

It says 95% of the posterior probability for the conversion rate falls between 1.9% and 2.5%. Given the model and prior you used, there is a 95% probability the rate lies in that range. The probability attaches to the parameter.

Probability fixes the coin's bias and varies the data; likelihood fixes the observed data and varies the bias. After 7 heads in 10 flips the likelihood is a curve over p that is not a probability distribution.

A count metric sums events, such as total orders. A rate metric divides a count by the units that were exposed, such as orders per visitor. A share metric divides one count by a related total, such as the percentage of orders placed on mobile.

A guardrail is a metric the experiment must not damage — latency, crash-free sessions, unsubscribes — and a big enough degradation can veto the launch. A driver metric only explains why the primary metric moved; it never vetoes.

Almost all the variance in revenue per user comes from a handful of very large spenders, so the interval around the treatment-control difference stays wide. Ordinary experiment sizes then cannot resolve realistic revenue effects.

The Overall Evaluation Criterion is the single metric, agreed before launch, that an experiment's ship-or-not decision is defined against. It expresses what the team means by the product getting better, so the result cannot be reinterpreted afterwards.

A DAU/MAU of 0.2 means the typical monthly active user opens the product on about 6 days out of 30. It measures return frequency, not audience size, and averages over very different user types.

It subtracts the comparison group's before-to-after change from the treated group's before-to-after change. That double difference cancels the fixed level gap between the groups and any shock that moved both of them over the same window.

An average pools opposite effects. A drug that raises recovery for patients under 50 and lowers it by a similar amount for patients over 70 posts an overall average near zero while both real effects remain untouched underneath it.

A propensity score is a unit's probability of receiving the treatment given its observed covariates. Conditioning on that single number balances the covariates that went into it, so matching happens in one dimension instead of many.

Adding a measured confounder turns the treatment coefficient from a raw comparison into a within-strata one: it now compares treated and untreated units that share the same covariate value, removing the part of the gap that covariate explained.

A sensitivity analysis says how strong an unmeasured confounder would have to be to overturn the estimate. It never shows confounding is absent; it prices how much hidden bias the finding can tolerate.

MAPE divides each absolute error by the actual value, so a zero-sales day makes that term undefined and a near-zero day makes it explode. On intermittent demand the average is dominated by a handful of tiny denominators.

A point outlier is one observation far from expectation, after which the series returns to its old level. A level shift moves the series to a new baseline that persists. The difference decides whether you drop a point or re-baseline.

A shuffled split trains on future days and tests on past ones, so the model interpolates between neighbouring dates instead of forecasting. Because adjacent days are highly correlated, the score looks excellent and says nothing about future performance.

Decomposition splits a series into three parts: trend-cycle, the slow movement of the level; seasonal, the pattern that repeats at a fixed known period such as 12 months; and remainder, the variation left after removing both.

Alpha sets how much weight the update puts on the newest observation versus the accumulated past. Alpha near 1 tracks recent data and reacts fast; alpha near 0 averages over a long history and smooths noise.